Relevance vector machine and multivariate adaptive regression spline for modelling ultimate capacity of pile foundation

Document Type : Research


1 Associate Professor, Centre for Disaster Mitigation and Management, VIT University, India,

2 Associate Professor, Department of Civil Engineering, Curtin University


This study examines the capability of the Relevance Vector Machine (RVM) and Multivariate Adaptive Regression Spline (MARS) for prediction of ultimate capacity of driven piles and drilled shafts. RVM is a sparse method for training generalized linear models, while MARS technique is basically an adaptive piece-wise regression approach. In this paper, pile capacity prediction models are developed based on data obtained from the literature and comprise in-situ pile loading tests and Cone Penetration Test (CPT) results. Equations are derived from the developed RVM and MARS models, and the prediction results are compared with those obtained from available CPT-based methods. Sensitivity has been carried out to determine the effect of each input parameter. This study confirms that the developed RVM and MARS models predict ultimate capacity of driven piles and drilled shafts reasonably well, and outperform the available methods.


1. Ardalan H, Eslami A, Nariman-Zadeh N. Pile shaft capacity from CPT and CPTu data by polynomial neural networks and genetic algorithms. Computers and Geotechnics 2008; 36: 616- 625. [DOI:10.1016/j.compgeo.2008.09.003]
2. Shahin MA Intelligent computing for modeling axial capacity of pile foundations. Canadian Geotechnical Journal 2010; 47: 230-243. [DOI:10.1139/T09-094]
3. Park D, Rilett, LR. Forecasting freeway link ravel times with a multi-layer feed forward neural network. Computer Aided Civil and Infrastructure Engineering 1999; 14: 358 - 367. [DOI:10.1111/0885-9507.00154]
4. Kecman V. Learning and Soft Computing: Support Vector Machines, Neural Networks, and Fuzzy Logic Models. The MIT Press:Cambridge, Massachusetts, London, England, 2001.
5. Tipping ME. The relevance vector machine. Advances in Neural Information Processing Systems 2000; 12: 625-658.
6. Friedman JH Multivariate adaptive regression splines. Ann Stat 1991; 19:1-141. [DOI:10.1214/aos/1176347973]
7. Brinch Hansen J. Discussion on hyperbolic stress-strain response, cohesive soils. Journal of Soil Mechanics and Foundation Engineering 1963; 89: 241-242.
8. Eslami A.Bearing capacity of piles from cone penetration test data. Ph.D. thesis, University of Ottawa, Ottawa, Ontario, 1996.
9. Alsamman OM. The use of CPT for calculating axial capacity of drilled shafts. Ph.D. thesis, University of Illinois-Champaign, Urbana, 1995.
10. Reese LC, O'Neil MW. Drilled shafts: Construction procedure and design methods. U.S. Department of Transportation, Dallas, Tex. Report FHWA-HI-88-042, 1988.
11. Tipping ME. Sparse Bayesian learning and the relevance vector machine. Journal of Machine Learning Research 2001; 1: 211-244.
12. Berger JO. Statistical Decision Theory and Bayesian Analysis, 2nd ed. Springer: New York, 1985. [DOI:10.1007/978-1-4757-4286-2]
13. Wahba G. A comparison of GCV and GML for choosing the smoothing parameters in the generalized spline-smoothing problem. Ann. Stat. 1985; 4: 1378-1402. [DOI:10.1214/aos/1176349743]
14. MacKay DJ. Bayesian methods for adaptive models. Ph.D. thesis, Department of Computing and Neural Systems. Calif Inst. of Technol., Pasadena. Calif,1992.
15. de Ruiter J, Beringern FL. Pile foundation for large North Sea structures. Marine Geotechnology 1979; 3: 267-314. [DOI:10.1080/10641197909379805]
16. Eslami A, Fellenius BH. Pile capacity by direct CPT and CPTu methods applied to 102 case histories. Canadian Geotechnical Journal 1997; 34: 886-904. [DOI:10.1139/t97-056]
17. Schmertmann, J.H. 1978. Guidelines for cone penetration test, performance and design. U.S. Department of Transportation, Washington, D.C. Report No FHWA-TS-78-209